9504004 7 Apr 1995 The Robinson-Schensted
The Robinson-Schensted correspondence as the quantum straightening at q = 0 Bernard
Leclercy and Jean-Yves Thibonz
q-alg/9504004 7 Apr 1995
March 1995.
Dedicated to Dominique Foata
Abstract
We show that the quantum straightening algorithm for Young tableaux and Young bitableaux reduces in the crystal limit q 7! 0 to the Robinson-Schensted algorithm.
1 Introduction Let K be a eld of characteristic 0, and X = (xij ) i; jn a matrix of commutative indeterminates. The ring K [xij ] may be regarded as the algebra F [Mat n] of polynomial functions on the space of n n matrices over K . A linear basis of this algebra is given by the bitableaux of Desarmenien, Kung, Rota [6], which are de ned in the following way. Given two (semistandard) Young tableaux and 0 of the same shape, with columns c ; : : :; ck and c0 ; : : :; c0k , the Young bitableau ( j 0) is the product of the k minors of X whose row indices belong to ci and column indices to c0i, i = 1; : : : ; k. For example, 1
1
1
4 3 5 6 1 1 2
5 2 4 7 1 3 5
x x x := x x x xx xx xx xx : x x x More generally, we shall call tabloid a sequence of column-shaped Young tableaux, and we shall associate to each pair ; 0 of tabloids of the same shape a bitabloid ( j 0) de ned as the product of minors indexed by the columns of and 0.
11
12
31
32
41
42
15 35 45
13
53
14 54
25
65
27 67
Partially supported by PRC Math-Info and EEC grant n0 ERBCHRXCT930400 L.I.T.P., Universite Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France Institut Gaspard Monge, Universite de Marne-la-Vallee, 2 rue de la Butte-Verte, 93166 Noisy-leGrand cedex, France y z
1
There exists an algorithm due to Desarmenien [5] for expanding any polynomial in K [xij ] on the basis of bitableaux. This is the so-called straightening algorithm (for bitableaux). In particular, the monomials xi1j1 xikjk , which obviously form another linear basis of K [xij ], can be expressed in a unique way as linear combinations of bitableaux. Thus, the straightening of x x x reads 23
2 1 3 2 + 1 3
11
3 1 2 3 3 1 2 + 1 2
32
= 1 2 3
1 2 3
2 2 1 3 ? 1 3
3
3 1 2
? 1 2 2 1 3
+
3 2 1
3 2 1
On the other hand, the Robinson-Schensted correspondence [25, 26, 27, 16] associates to any word w on the alphabet of symbols f1; : : : ; ng a pair (P (w); Q(w)) of Young tableaux of the same shape. For example, the image of the word w = 2 1 4 3 5 1 2 under this correspondence is the pair 4 2 3 5 1 1 2 ,
6 2 4 7 1 3 5
Both the straightening algorithm and the Robinson-Schensted algorithm are strongly connected with the representation theory of GLn . Indeed, the straightening algorithm allows to compute the action of GLn in (polynomial) irreducible representations, while the Robinson-Schensted correspondence was devised by Robinson to obtain a proof of the Littlewood-Richardson rule for decomposing into irreducibles the tensor product of two irreducible representations. The problem that we want to investigate is whether there exists any relation between the straightening algorithm and the Schensted algorithm. It turns out that to answer this question, one has to replace the algebra F [Mat n ] by its quantum analogue Fq [Mat n] [24]. This is the associative algebra over K (q) generated by n letters tij ; i; j = 1; : : :; n subject to the relations 2
tik til tik tjk til tjk tik tjl ? tjl tik for 1 i < j n and 1 k < l n. element det q = det q T :=
q? til tik ; (1) ? q tjk tik ; (2) tjk til ; (3) ? (q ? q) til tjk ; (4) The quantum determinant of T = (tij ) is the
= = = =
X
1
1
1
(?q)?` w t w1 : : : tnwn ( )
w2Sn
2
1
of Fq [Matn ]. Here Sn is the symmetric group on f1; : : : ; ng and `(w) denotes the length of the permutation w. The quantum determinant of T belongs to the center of Fq [Matn ]. More generally, for I = (i ; : : : ; ik) and J = (j ; : : :; jk ) one de nes the quantum minor of the submatrix TIJ as 1
1
det q TIJ :=
X
(?q)?` w ti1jw1 : : : tikjwk ; ( )
w2Sk
and the quantum bitableau ( j 0) (resp. quantum bitabloid ( j 0)) as the product of the quantum minors indexed by the columns of the Young tableaux and 0 (resp. of the tabloids and 0). As proved by Huang, Zhang [10], the set of quantum bitableaux is again a linear basis of Fq [Mat n]. For example, the expansion of the monomial t t t on this basis is 23 11 32
2 1 3
2 + (1 ? q + q ) 1 3 2
= q
3 1 2
3 1 2
4
3
1 2 3
1 2 3
3 +q 1 2
2 1 3
4
+q
3 2 1
5
3 ?q 1 2
3 1 2
2 1 3
2 1 3
3
? q
5
3 2 1
In the present example the coecients of the expansion are polynomials in q, and only one of them has a nonzero constant term. In other words, denote by B the linear basis of quantum bitableaux in Fq [Mat n], and let L be the lattice generated over K [q] by the elements of B. Then 2 3 1 2 1 3
2 1 3
3 1 2
mod qL
This is an illustration of the following Theorem 1.1 Let w = i ik and u = j jk be two words on f1; : : : ; ng, and denote by (P (w); Q(w)), (P (u); Q(u)) their images under the Robinson-Schensted correspondence. Then, qL if Q(w) = Q(u) : ti1j1 tik jk (P (w)0jP (u)) mod mod qL otherwise 1
1
3
As a corollary we obtain an unexpected characterization of the plactic congruence on words [16, 19], de ned by
w u () P (w) = P (u) :
Corollary 1.2 With the same notations as above, w u () ti1i1 tikik tj1j1 tjk jk mod qL : At this point, it is important to recall that the existence of a connection between the Robinson-Schensted correspondence and the representation theory of the quantized enveloping algebra Uq (gl n ) at q = 0 was rst discovered by Date, Jimbo, Miwa [4]. The results presented here are in fact of the same kind as those of [4], namely, it is shown in [4] that if V denotes the basic representation of Uq (gl n ), the transition matrix in V k from the basis of monomial tensors to the Gelfand-Zetlin basis specializes when q = 0 to a permutation matrix given by the Robinson-Schensted map. Similarly, Theorem 1.1 states that the transition matrix in the Uq (gl n )-module Fq [Mat n] from the basis of monomials (1)
(1)
B = fti1j1 tik jk j Q(i ik) = Q(j jk ) = for some g to the basis of quantum bitableaux is equal at q = 0 to a permutation matrix also computed from the Robinson-Schensted algorithm. Here, denotes for each partition a xed standard Young tableau of shape . The work of Date, Jimbo, Miwa, provided the starting point from which Kashiwara developed his theory of crystal bases for quantized enveloping algebras [13, 14]. We shall use crystal bases as the main tool for proving Theorem 1.1. The paper is organized as follows. In Section 2, we collect the necessary material about Uq (gl n), Uq (sl n) and their representation theory. In Section 3, we review the de nition and basic properties of Kashiwara's crystal bases at q = 0. In Section 4, we formulate and prove a version of Theorem 1.1 for the case of (single) tableaux, that is, we work in the subring of Fq[Mat n ] generated by the quantum minors taken on the initial rows of T . Finally in Section 5 we prove Theorem 1.1, as well as a slightly more general statement. 1
2
Uq (gl n)
1
and Uq(sl n)
A general reference for this Section and the following one is the excellent exposition [3]. We rst recall the de nition of the quantized enveloping algebras Uq (gl n ) [12] and Uq (sl n) [11, 7]. Uq (gl n) is the associative algebra over K (q) generated by the 4n ? 2 symbols ei; fi; i = 1; : : :; n ? 1 and qi ; q?i ; i = 1; : : :; n, subject to the relations
qi q?i = q?i qi = 1 ; [qi ; qj ] = 0 ; qej for i = j qi ej q?i = q? ej for i = j + 1 ; ej otherwise 8 >
:
4
(5) (6)
q? fj for i = j (7) qi fj q?i = qfj for i = j + 1 ; fj otherwise i ?i+1 ?i i+1 [ei; fj ] = ij q q q ??qq? q ; (8) [ei; ej ] = [fi; fj ] = 0 for ji ? j j > 1 ; (9) ej ei ? (q + q? )eiej ei + ei ej = fj fi ? (q + q? )fifj fi + fi fj = 0 for ji ? j j = 1 : (10) The subalgebra of Uq (gl n) generated by ei; fi, and qhi = qi q?i+1 ; q?hi = q?i qi+1 ; i = 1; : : : ; n ? 1; (11) is denoted by Uq (sl n). The representation theories of Uq (gl n ) and Uq (sl n) are closely parallel to those of their classical counterparts U (gl n) and U (sl n ). Let M be a Uq (gl n)-module and = ( ; : : : ; n) be a n-tuple of nonnegative integers. The subspace M = fv 2 M j qi v = qi v ; i = 1; : : : ; ng is called a weight space and its elements are called weight vectors (of weight ). Relations (6) (7) show that eiM M+ ; fiM M? ; where = ( ; : : :; i + 1; i ? 1; : : : ; n ) and ? = ( ; : : :; i ? 1; i + 1; : : : ; n). Thus, ordering the weights in the usual way by setting 8 >
:
1
2
1
2
2
1
2
1
+
1
+1
()
1
k
X
i=1
i
k
X
i=1
i ;
+1
k = 1; : : : ; n;
we see that the ei's act as raising operators and the fi's as lowering operators. A weight vector is said to be a highest weight vector if it is annihilated by the ei's. M is called a highest weight module if it contains a highest weight vector v such that M = Uq (gl n ) v. If v is of weight , it follows that dim M = 1 and M = M . One then shows that there exists for each partition of length n a unique highest weight nite-dimensional irreducible Uq (gl n)-module V, with highest weight .
Example 2.1 The basic representation V = V of Uq (gl n ) is the n-dimensional vector space over K (q) with basis fvi; 1 i ng, on which the action of Uq (gl n ) is as follows: (1)
qi vj = qij vj ; eivj = i j vi; fi vj = ij vi : +1
+1
Example 2.2 More generally, the Uq (gl n )-module V k is a nk -dimensional vector space with basis fvcg labelled by the subsets c of f1; : : : ; ng with k elements (i.e. by the Young tableaux of shape (1k ) over f1; : : : ; ng). The action of Uq (gl n) on this basis is given by qi vc = vc if i 62 c ;
(1 )
qvc otherwise 5
i + 1 62 c or i 2 c eivc = v0 otherwise; ifwhere d = (c n fi + 1g) [ fig ; d i + 1 2 c or i 62 c fivc = v0 otherwise; ifwhere d = (c n fig) [ fi + 1g : d We see that the action of the lowering operators fi does not depend on q, and can be recorded on a colored graph whose vertices are the column-shaped Young tableaux c and whose arrows are given by: i c ?! d () fivc = vd : Thus for k = 2; n = 4, one has the following graph: 2 2 3 3 4 1 1 1
1 3 2
1 3
4 2 2 4 3
This is one of the simplest examples of crystal graphs (cf. Section 3). In order to construct more interesting Uq (gl n )-modules, we use the tensor product operation. Given two Uq (gl n )-modules M; N , we can de ne a structure of Uq (gl n )module on M N by putting qi (u v) = qi u qi v; (12) ? h i ei(u v) = eiu v + q u eiv; (13) h i fi(u v) = fiu q v + u fiv: (14) Indeed, the formulas qi = qi qi ; ei = ei 1 + q?hi ei; fi = fi qhi + 1 fi; de ne a comultiplication on Uq (gl n). One shows that the decomposition into irreducible components of the tensor product of two irreducible Uq (gl n )-modules is given by V V ' c V ; (15) M
where the c are the classical Littlewood-Richardson numbers. In particular, it follows that V k ' f V ; (16) M
`k
where f denotes the number of standard Young tableaux of shape . 6
Example 2.3 The n -dimensional Uq (gl n )-module V decomposes into the q-symmetric 2
2
square V and the q-alternating square V by the following diagram: (2)
;
(1 1)
. For n = 2 this decomposition is described
f1 f1 f1 0 e1? v v ?! v v + qv v ?! (q + q? ) v v ?! 0 'V 1
1
1
2
2
1
1
2
f1 0 e1? v v ? qv v ?! 0 'V 2
1
1
2
2
(2)
;
(1 1)
The algebra Fq [Mat n] de ned in Section 1 is also endowed with a natural structure of Uq (gl n)-module via the action de ned by
qi tkl = qil tkl; ei tkl = i l tk l? ; fi tkl = il tk l ; (17) and the Leibniz formulas (18) qi (PQ) = (qi P ) : (qi Q); ? h i ei(PQ) = (eiP ) : Q + (q P ) : (eiQ); (19) h (20) fi(PQ) = (fiP ) : (q i Q) + P : (fiQ); for P; Q in Fq [Mat n]. This provides a very convenient realization of the irreducible modules V as natural subspaces of Fq [Mat n]. To describe it, we introduce some notations. We shall write y for the unique Young tableau of shape and weight . This is the socalled Yamanouchi tableau of shape . Let be any Young tableau of shape . The quantum bitableau (y j ) will be simply denoted by ( ) and will be called a quantum tableau. This is a product of quantum minors taken on the rst rows of the matrix T . Quantum tabloids are de ned similarly. Finally, denote by T the subspace of Fq [Mat n ] spanned by quantum tableaux ( ) of shape . Then one can show [18, 22] the following q-analogue of a classical result of Deruyts (see [9]). Theorem 2.4 The subspace T is invariant under the action of Uq (gl n) on Fq [Mat n ], and is isomorphic as a Uq (gl n)-module to the simple module V . The action of Uq (gl n ) on T is computed by means of the q-straightening formula. Namely, for column-shaped quantum tableaux one checks easily that the action coincides with the one previously described in Example 2.2. For general quantum tableaux we use Leibniz formulas (18) (19) (20), and when necessary we use the q-straightening algorithm (cf. Section 4) for converting the quantum tabloids of the right-hand side into a linear combination of quantum tableaux. Example 2.5 We choose n = 3 and = (2; 1). T ; is 8-dimensional and one has for instance 3 3 3 3 2 f 1 1 = 1 2 + q 2 1 = (1 + q ) 1 2 ?q 1 3 +1
1
+1
(2 1)
2
1
3
In this example, the quantum tableaux ( ) have been written for short (without brackets). This small abuse of notation will be used freely in the sequel. 7
This realization of V is, up to some minor changes of convention, the same as the one described in [2] via the q-Young symmetrizers of the Hecke algebra of type A. We denote by Fq[GLn =B ] the subspace of Fq [Mat n] spanned by the quantum tableaux. It follows from the q-straightening formula that this is in fact a subalgebra of Fq[Mat n ]. It coincides for q = 1 with the ring of polynomial functions on the ag variety GLn =B , hence the notation. The quantum deformation Fq [GLn =B ] has been studied by Lakshmibai, Reshetikhin [18] and Taft, Towber [28]. As a Uq (gl n )-module it decomposes into:
Fq [GLn =B ] '
M
`()n
V :
(21)
Returning to Fq [Mat n ] and its linear basis formed by quantum bitableaux, we note that the action of Uq (gl n) de ned by (17) involves only the column indices of the variables tij . The de ning relations (1) (2) (3) (4) being invariant under transposition of the matrix T , we see that we have another action of Uq (gl n ) given by (qi )ytkl = qik tkl; eyi tkl = i
fiytkl = ik tk l; (22) (where the symbol y has been added to distinguish this action from the previous one), and the Leibniz formulas (18) (19) (20). These two actions obviously commute with each other, so that Fq [Mat n] is now endowed with the structure of a (left) bimodule over Uq (gl n). The quantum version of the Peter-Weyl theorem provides the decomposition [22] Fq [Mat n ] ' V V : (23) +1
k tk?1 l ;
+1
M
`()n
Here, the irreducible bimodule V V is generated by applying all possible products of lowering operators fiy; fj to the highest weight vector (yjy). We end this Section by noting that every Uq (gl n)-module M can be regarded by restriction as a Uq (sl n )-module (that we still denote by M ). In particular, the V are also irreducible under Uq (sl n ). However we point out that, as Uq (sl n)-modules,
V ' V () i ? i = i ? i ; i = 1; : : : ; n ? 1 : +1
+1
Example 2.6 The Uq (sl )-modules V l will be very important in the sequel and we describe them precisely. For l 0, V l is a (l + 1)-dimensional vector space over K (q) with basis fuk ; 0 k lg, on which the action of Uq (sl ) is as follows: qh1 uk = ql? k uk ; e uk = [l ? k + 1] uk? ; f uk = [k + 1] uk : In these formulas [m] denotes the q-integer (qm ? q?m)=(q ? q? ), and we understand u? = ul = 0. Setting [m]! = [m][m ? 1] [1] and f m = f m =[m]!, we see that the basis fuk g is characterized by uk = f k u . Also, we note that the weight spaces being one-dimensional, there is up to normalization a unique basis of V l whose elements are weight vectors. The basis fuk g may therefore be regarded as canonical. This will provide 2
()
()
2
2
1
1
1
+1
1
1
+1
( ) 1
( 1
)
1
0
()
the starting point for de ning the crystal basis of a Uq (gl n )-module. 8
3 Crystal bases
It follows from relations (6) (7) (8) (11) that for any i = 1; : : :; n ? 1, the subalgebra Ui generated by ei; fi; qhi ; q?hi is isomorphic to Uq (sl ). Hence a Uq (gl n )-module M can be regarded by restriction to Ui as a Uq (sl )-module. We shall assume from now on that the weight spaces M are nite-dimensional, that M = M , and that for any i, M decomposes into a direct sum of nite-dimensional Ui-modules. Such modules M are said to be integrable. It follows from the representation theory of Uq (sl ) that for any i, the integrable module M is a direct sum of irreducible Ui-modules V l . Example 3.1 Let M denote the Uq (gl )-module V ; in the realization given by Theorem 2.4. As a U -module, M decomposes into 4 irreducible components, as shown by the following diagram: 2
2
2
()
3
(2 1)
1
0
e
1
0
e
1
0
e
1
0
e
1
2 1 1
f
3 1 1
f
2 1 3
f
3 1 3
f
2 1 2
1
f
0
1
3 (1 + q ) 1 2 ? q
1
2
3
2 1 3
f =[2] 1
3 2 2
f
0
1
0
1
3 2 3
1
f
0
1
On the other hand, as a U -module, M decomposes into: 2
0
e
2 1 1
f
0
e
2 1 2
f
0
e
3 1 2 ?q
0
e
3 2 2
2
2
2
2
3 1 1
2
2
2
2 1 3 +q
2
2 1 3
f
f
f
0 3 1 2
0
2
3 2 3
f 9
2
0
f =[2] 2
3 1 3
f
2
0
We observe that the U -decomposition leads to the basis 1
2 B =( 1 3 + q 2
for the 2-dimensional weight space M 2 B =( 1 3 + q 2
3 1 2 ;;
(1 1 1)
3 ; 1 2 ?q
2 1 3 )
, while the U -decomposition leads to 2
3 1 2
3 ; 1 2 ?q
2 1 3 )
These two bases are di erent and therefore one cannot nd a basis B of weight vectors in M compatible with both decompositions. However, as noted by Kashiwara, `at q = 0' the bases B and B coincide. The next de nitions will allow us to state this in a more formal way. Consider the simple Uq (sl )-module V l with basis fuk g (cf. Example 2.6). Kashiwara [13, 14] introduces the endomorphisms e~; f~ of V l de ned by e~ uk = uk? ; f~ uk = uk ; k = 0; : : : ; l; 1
2
2
()
()
1
+1
where u? = ul = 0. More generally, if M is a direct sum of modules V , that is, if there V l , one de nesl endomorphisms exists an isomorphism of Uq (sl )-modules : M ?! l ~ e~; f of M by means of in the obvious way, and one checks easily that they do not depend on the choice of . In particular, if M is an integrable Uq (gl n )-module, regarding M as a Ui -module we de ne operators e~i; f~i on M for i = 1; : : : ; n ? 1. Example 3.2 We keep the notations of Example 3.1. We have 2 3 3 f~ 1 3 +q 1 2 1 3 1
+1
()
2
()
2
3 2 1 2 ?q 1 3
f~
0
2
Hence, 2 1 3
1 31 3 1+q 3 q 31 3 f~ 1 2 1+q Since we want to let q tend to 0, we introduce the subring A of K (q) consisting of rational functions without pole at q = 0. A crystal lattice of M is a free A-module L such that M = K (q) A L; L = L where L = L \ M, and e~iL L ; f~iL L ; i = 1; : : : ; n ? 1 : (24)
f~
2
2
2
2
10
In other words, L spans M over K (q), L is compatible with the weight space decomposition of M and is stable under the operators e~i and f~i. It follows that e~i; f~i induce endomorphisms of the K -vector space L=qL that we shall still denote by e~i; f~i. Now Kashiwara de nes a crystal basis of M (at q = 0) to be a pair (L; B ) where L is a crystal lattice in M and B is a basis of L=qL such that B = tB where B = B \ (L =qL), and e~iB B t f0g ; f~iB B t f0g ; i = 1; : : : ; n ? 1 ; (25)
e~iv = u () f~iu = v; u; v 2 B; i = 1; : : : ; n ? 1 :
(26)
Example 3.3 We continue Examples 3.1 and 3.2. Denote by B the basis of quantum tableaux in M , and let L be the A-lattice in M spanned by the elements of B. Let B be the projection of B in L=qL. Then, Example 3.1 shows that (L; B ) is a crystal basis of M.
Kashiwara has proven the following existence and uniqueness result for crystal bases [13, 14]. Theorem 3.4 Any integrable Uq (gl n )-module M has a crystal basis (L; B ). Moreover, if (L0 ; B 0) is another crystal basis of M , then there exists a Uq (gl n )-automorphism of M sending L on L0 and inducing an isomorphism of vector spaces from L=qL to L0 =qL0 which sends B on B 0 . In particular, if M = V is irreducible, its crystal basis (L(); B ()) is unique up to an overall scalar multiple. It is given by A f~i1 f~i2 f~ir u; (27) L() = X
i1 ;i2 ;:::;ir n?1
1
B () = ff~i1 f~i2 f~ir u mod qL() j 1 i ; : : : ; ir ngnf0g; (28) where u is a highest weight vector of V . It follows that one can associate to each integrable Uq (gl n )-module M a well-de ned colored graph ?(M ) whose vertices are labelled by the elements of B and whose edges describe the action of the operators f~i : 1
i u ?! v () f~iu = v : ?(M ) is called the crystal graph of M . Example 3.5 The crystal graph of the Uq (gl )-module V ; is readily deduced from Examples 3.1, 3.2 and 3.3. It is shown in Figure 1. It is clear from this de nition that the crystal graph ?(M ) of the direct sum M = M M of two Uq (gl n )-modules is the disjoint union of ?(M ) and ?(M ). It follows from the complete reducibility of M that the connected components of ?(M ) are the crystal graphs of the irreducible components of M . Note also that if one restricts the crystal graph ?(M ) to its edges of colour i, one obtains a decomposition of this graph into strings of colour i corresponding to the Ui-decomposition of M . For a vertex v of 3
1
2
(2 1)
1
11
2
2 11
3 11
2
3 12
1
1
1
2 12
2
2 13
3 22
2
2
3 13
3 23
1
Figure 1: Crystal graph of the Uq (gl )-module V 3
;
(2 1)
?(M ), we shall denote by i(v) (resp. i(v)) the distance of v to the origin (resp. end) of its string of colour i, that is, i(v) = maxfk j e~ki v 6= 0g ; i(v) = maxfk j f~ik v 6= 0g : The integers i(v), i(v) give information on the geometry of the graph ?(M ) around the vertex v. They seem to be very signi cant from a representation theoretical point of view. Indeed, both the Littlewood-Richardson multiplicities c and the q-weight multiplicities K (q) can be computed in a simple way from the integers i(v), i(v) attached to the crystal graph ?(V ) [1, 17]. One of the nicest properties of crystal bases is that they behave well under the tensor product operation. We shall rst consider an example, from which Kashiwara deduces by induction the general description of the crystal basis of a tensor product [13].
Example 3.6 We slightly modify the notations of Example 2.6 and write fukl ; k = 0; : : : ; lg for the canonical basis of the Uq (sl )-module V l . We recall that for convenience we set u?l = ul l = 0. A rst basis of the tensor product V V l is fuj ukl ; j = 0; 1; k = 0; : : :; lg. We de ne wk = u ukl + ql?k u ukl? ; k = 0; : : :; l + 1 ; (29) l l zk = ql?k? [l ? k] u uk ? ql [k + 1] u uk ; k = 0; : : : ; l ? 1 : (30) ()
2
() 1
()
() +1
(1)
(1) 0
()
+1
(1) 1
1
(1) 1
()
(1)
()
1
()
(1) 0
() +1
It is straightforward to check that e w = e z = 0; f wl = f zl? = 0, and 1
0
1 0
1
+1
1
f wk = [k + 1] wk ; f zk = [k + 1] zk : 1
+1
1
12
+1
1
()
Hence fwk g and fzk g span submodules of V V l isomorphic to V l and V l? respectively, and are the canonical bases of these irreducible representations of Uq (sl ). Therefore, the A-lattice L spanned by fwk g, fzk g is a crystal lattice and if we de ne B = fwk mod qLgtfzk mod qLg, then (B; L) is a crystal basis of V V l . On the other hand, equations (29) (30) show that L coincides with the tensor product L(1) L(l) of the crystal lattices of V and V l , and that (1)
()
( +1)
(
1)
2
(1)
(1)
()
()
wk u ukl mod qL ; k = 0; : : : ; l ; wl u ul l mod qL : (1) 0
()
(1) 1
+1
()
zk u ukl mod qL ; k = 0; : : : ; l ? 1 : Thus, (L; B ) coincides with (L(1) L(l); B (1) B (l)), where we have denoted by B (1)
B (l) the basis fuj ukl ; mod qL ; j = 0; 1; k = 0; : : : ; lg. Finally, the action of f~ on B (1) B (l) is described by the crystal graph: (1) 1
(1)
()
()
1
u u l ! u u l ! ! u ul?l ! u ul l (1) 0
() 0
(1) 0
() 1
(1) 0
() 1
(1) 0
()
# l u ul
u u ! u u ! ! u ul? More generally, we have the following property [13]. Theorem 3.7 Let (L ; B ) and (L ; B ) be crystal bases of integrable Uq (gl n )-modules M and M . Let B B denote the basis fu v; u 2 B ; v 2 B g of (L =qL ) (L =qL ) (which is isomorphic to (L L )=q(L L )). Then, (L L ; B B ) is a crystal basis of M M , the action of e~i, f~i on B B being given by ~ i(v) ; f~i(u v) = uf~ u
fivv ifif i((uu)) < (31) i(v) i i > i(v) : e~i(u v) = eu~iu
e~ vv ifif i((uu)) (32) (v) l
(1) 1
() 0
1
2
1
l
(1) 1
() 1
1
2
l
() 1
(1) 1
()
2
1
2
1
1
1
(1) 1
2
1
2
2
2
1
1
1
2
1
1
2
2
2
2
i
i
i
Theorem 3.7 enables one to describe the crystal graph of V m for any m, and to deduce from that the description of the crystal graph ? of the simple Uq (gl n )-module V. For convenience, we shall identify the tensor algebra T (V ) with the free associative algebra K (q)hAi over the alphabet A = f1; : : : ; ng via the isomorphism vi 7! i, where fvig is the canonical basis of V de ned in Example 2.1. Accordingly, the crystal lattice L spanned by the monomials in the vi is identi ed with AhAi and the K -vector space L=qL with K hAi. De ne linear operators e^i, f^i on K hAi in the following way. Consider rst the case of a two-letter alphabet A = fi; i+1g. Let w = x xm be a word on A. Delete every factor ((i+1) i) of w. The remaining letters constitute a subword w of w. Then, delete again every factor ((i+1) i) of w . There remains a subword w . Continue this procedure until it stops, leaving a word wk = xj1 xjr+s of the form wk = ir(i+1)s. The image of wk under e^i; f^i is the word yj1 yjr+s given by (1)
(1)
(1)
1
1
1
e^i(ir(i+1)s ) =
ir (i+1)s? (s 1) ; 0 (s = 0) +1
2
r? s f^i(ir(i+1)s ) = i (i+1) 0
1
13
1
+1
(r 1) : (r = 0)
The image of the initial word w is then w0 = y ym, where yk = xk for k 62 fj ; : : :; jr s g. For instance, if w = (2 1) 1 1 2 (2 1) 1 1 1 2 ; we shall have w = : : 1 1 (2 : : 1) 1 1 2 ; w = : : 1 1 : : : : 1 1 2: Thus, e^ (w) = 2 1 1 1 2 2 1 1 1 1 1 ; f^ (w) = 2 1 1 1 2 2 1 1 1 2 2 ; where the letters printed in bold type are those of the image of the subword w . Finally, in the general case, the action of the operators e^i; f^i on w is de ned by the previous rules applied to the subword consisting of the letters i; i+1, the remaining letters being unchanged. The operators e^i, f^i have been considered in [20], where they were used as building blocks for de ning noncommutative analogues of Demazure symmetrization operators. It follows from Theorem 3.7 that they coincide in the above identi cation with the endomorphisms e~i, f~i on the K -space L=qL [15]. It is straightforward to deduce from the de nition of e^i, f^i the following compatibility properties with the Robinson-Schensted correspondence: (a) for any word w on A such that e^i w 6= 0, we have Q(^ei w) = Q(w), (b) for any pair of words w, u such that P (w) = P (u) and e^iw 6= 0, we have P (^eiw) = P (^eiu). In other words, the operators e^i, f^i do not change the insertion tableau and are compatible with the plactic equivalence. Moreover, the words y such that e^iy = 0 for any i are characterized by the Yamanouchi property: each right factor of y contains at least as many letters i than i +1, and this for any i. It is well-known that for any standard Young tableau there exists a unique Yamanouchi word y such that Q(y) = . This yields the following cristallization of (16). Theorem 3.8 The crystal graph of the Uq (gl n)-module V m is the colored graph whose vertices are the words of length m over A, and whose edges are given by: i w ?! u () f^iw = u : The connected components ? of ?(V m ) are parametrized by the set of Young tableaux of weight (1m ). The vertices of ? are those words w which satisfy Q(w) = . Moreover, if denotes the shape of , then ? is isomorphic to the crystal graph ?(V). Replacing the vertices w of ? by their associated Young tableaux P (w), we obtain a labelling of ?(V) by the set of Young tableaux of shape , as shown in Example 3.5. It follows from property (b) above that this labelling does not depend on the particular choice of among the standard Young tableaux of shape . We end this section by showing a less trivial example of crystal graph, which is computed easily using the previous description of f^i. Example 3.9 The crystal graph of the Uq (gl )-module V ; is shown in Figure 2. 1
1
+
1
2
1
1
2
(1)
(1)
4
14
(2 2)
44 11
1
3
44 12
1
3 24 11
1
2
3 22 11
2
23 11
34 1 11
24 12
2 3
1
2
3
23 12
33 11 2
3 1
33 12
2 44 22
44 13 2
1
44 23
3
2
44 33
3 34 3 12 1 34 13 1 24 2 13 34 2 34 22 23 3 1
33 22
Figure 2: Crystal graph of the Uq (gl )-module V 4
4 Fq[GLn=B ]
;
(2 2)
The subalgebra Fq[GLn =B ] of Fq[Matn] generated by quantum column-shaped tableaux can also be de ned by generators and relations [28, 18]. The generators are denoted by columns ik ... 1 k n; 1 i ; : : : ; ik n; i i and their products are written by juxtaposition. The relations are 1
2 1
15
(R ) 1
if ir = is for some indices r; s, then
ik ... = 0; i i 2 1
(R ) 2
for w 2 Sk and i < i < < ik , we have 1
2
iwk ... = (?q)?` w iw2 iw1
( )
(R ) 3
ik ... ; i i 2 1
for k l and j < j < < jl, we have 1
2
jwl ...
jl ...
jk j. k = .. j +1
i. k .. i
(?q)`(w)
X
w
1
1
1
jwk+1 i.k .. i
jw. k .. jw1
where the sum runs through all w 2 Sl such that w < < wk and wk < < wl, (R ) for k + s l ? 1 and j < j < < jl, we have 1
4
1
+1
2
(?q)` w
X
w
( )
jwl ... jwk+2 jwk+1 ik ? ... i 1
1
rs ... r jwk = 0 jwk?1 ... jw1 1
where the sum runs through all permutations w 2 Sl such that w < < wk and wk < < wl. Relations (R ), (R ) express the q-alternating property of quantum minors, while relations (R ), (R ) are the natural q-analogues of Sylvester's and Garnir's identities respectively (cf. [21]). Using (R ), (R ), (R ) one can express every element of Fq[GLn =B ] as a linear 1
+1
1
3
2
4
1
2
3
16
combination of products of columns, the size of which weakly decreases from left to right. After that, following the same strategy as in the classical straightening algorithm, one can by means of (R ) express all such products as linear combinations of quantum tableaux. Note that it follows from (R ), (R ) that all the quantum tableaux ( ) appearing in the straightening of a given tabloid () have the same shape obtained by reordering the sizes of the columns of , which means that () falls in the irreducible component V of Fq [GLn =B ]. Example 4.1 We apply the q-straightening algorithm to 4
3
4
6 () = 51 32 It follows from (R ), (R ), (R ) that 1
2
3
6 6 5 5 5 3 = 5 3 + 3 6 ?q 2 6 1 2 1 2 1 2 1 3 Using (R ) for straightening the rst term of the right-hand side, we obtain 4
6 5 3 =(1 ? q 1 2
2
5 5 3 6 ) +(q ? q) 2 6 1 2 1 3
6 +q 3 5 ?q 1 2
3
2
6 2 5 1 3
3 ?q 2 6 1 5 4
Letting q tend to 0 in this expansion, we get an illustration of the next Theorem, which can be regarded as a version of Theorem 1.1 for Fq [GLn =B ].
Theorem 4.2 Let B denote the linear basis of Fq [GLn=B ] consisting of quantum tableaux, and let L be the K [q]-lattice generated by B. Let be a partition, and () a quantum tabloid
in V. Denote by w the word obtained by reading the columns of from top to bottom and left to right. Then, qL if P (w ) has shape : () (P (0w)) mod mod qL otherwise
The proof of Theorem 4.2 is derived from the following result which describes several interesting bases of V, and states that each of them gives rise to the same crystal basis at q = 0. We rst introduce some notations. Let be a partition, 0 = (0 ; : : : ; 0r ) its conjugate, and (0) = (01 ; : : :; 0r ) the composition obtained by permuting the parts of 0 using the permutation 2 Sr . A tabloid is said to have shape (0) if it is a sequence of column-shaped Young tableaux (c ; : : : ; cr), the size of the column ci being equal to 0i , i = 1; : : :; r. The column reading of is the word u obtained by reading the columns of from top to bottom and left to right. 1
1
17
Theorem 4.3 For any 2 Sr, the set of quantum tabloids B = f() j has shape (0) and P (u ) has shape g is a linear basis of V. The A-lattice L spanned by B is independent of as well as the projection B of B in L=qL, and (L; B ) is a crystal basis of V. For () 2 B and ( ) 2 B , we have () ( ) mod qL () P (u ) = P (u ) : Finally, if () is such that P (u ) has not shape , then () 0 mod qL. Proof: We x 2 Sr and we write = (1 ; : : :; r ) := (0). Recall from Example 2.2 that the q-analogue V(1k ) of the k th exterior power of the basic representation has a natural basis labelled by column-shaped Young tableaux. We introduce the K (q)-linear map ':
V
1
(1 )
V
2
(1 )
V
?! V
(1r )
b. d... .. ..
a c.
f ... ?! e
b. .. a
d. .. ... f... e c
Comparing (12) (13) (14) to (18) (19) (20) we see immediately that ' commutes with the action of Uq (gl n ) and hence is in fact a morphism of Uq (gl n )-modules. The image V is irreducible and by (15) it appears with multiplicity one in V 1 V r . Therefore, the restriction of ' to the irreducible component of V 1 V r with highest weight vector the tensor product y := (y 1 ) (y r ) of all highest weight vectors in V 1 ; : : :; V r is an isomorphism, and the kernel of ' is the sum of all components V ; 6= appearing in V 1 V r . We now turn to the crystal bases of these various Uq (gl n)-modules. It follows from Example 2.2 and Theorem 3.4 that, if we choose u k = y k , L(1k ) is the A-lattice spanned by the basis of column-shaped quantum tableaux, while B (1k ) is the projection of this basis in L(1k )=qL(1k ). Using Theorem 3.7 we deduce that a crystal basis of V 1 V r can be constructed by tensoring the crystal bases of the factors. Denote this basis by (L; B ). Then '(L ) is a crystal lattice in V. According to Theorem 3.4 such a lattice is unique up to an overall scalar multiple which can be determined by considering the highest weight space. Here we have (L) = A y and ' (y) = (y). Therefore, '(L ) is the crystal lattice L of V speci ed by L = A (y), and it does not depend on . Now ' induces an isomorphism from the K -subspace of L=qL spanned by the subset of B consisting of those b in the connected component of the crystal graph with source y mod qL, to the K -space L=qL. It is easy to check from Theorem 3.7 and the explicit description of the operators e^i, f^i which follows it that this connected component is labelled by the elements b = c cr mod L such that the word (1
(1
(1
(1
)
)
(1
)
(1
(1
)
1
18
(1 )
)
(1
(1
)
(1 )
)
)
)
)
(1
(1
(1
)
)
ub obtained by reading c from top to bottom, then c from top to bottom, and so on, satis es: P (ub) has shape . Hence its image is B mod qL. This proves that B is a basis of V and that L = b2B A b. Moreover, (L; B mod qL) is a crystal basis, which proves by unicity that B mod qL does not depend on . On the other hand, the elements b of B which belong to the other connected components are sent to 0 in L=qL. This means that the tabloids () such that P (u ) has not shape belong to qL. Finally, if is another permutation and a tabloid of shape (0), the fact that () ( ) mod qL means exactly that the words u and u label the same vertex in the two copies of ?(V ) to which they belong in ?(V k ), that is, by Theorem 3.8, P (u ) = P (u ). 1
2
P
(1)
2
Proof of Theorem 4.2: Let () be a quantum tabloid in V. The q-straightening algorithm shows that () is a K [q; q?1]-linear combination of quantum tableaux. On the other hand, Theorem 4.3 shows that () belongs to the A-lattice L spanned by the set B = Bid of quantum tableaux. Therefore, () is in fact an element of the K [q]-lattice L spanned by B. The other statements of Theorem 4.2 follow immediately from Theorem 4.3. 2
We point out that the quantum tabloids () 2 B are easily computed using Schutzenberger's jeu de taquin [27]. For example, the following graph shows the 's giving rise to quantum tabloids () which are congruent to the quantum tableau ( ) modulo qL:
1
3 4 1 3 2 5
2
3 1 4 5 3 2
1
4 = 3 3 1 2 5
3 4 5 1 3 2 2
4 3 1 3 5 2
1
3 4 3 1 5 2
2
The edges labelled i connect tabloids ; obtained one from the other by permuting the column lengths of the i th and (i+1) th columns by means of jeu de taquin. The column readings u of these tabloids are distinguished words of the plactic class of u , called frank words [20]. The combinatorics of frank words occurs in several interesting problems (see [20, 23, 8]).
19
5 Proof of Theorem 1.1 The proof of Theorem 1.1 follows the same lines as that of Theorem 4.2, namely we rst describe several realizations of the crystal basis of the bimodule Fq [Matn] and then deduce Theorem 1.1 by comparing them, using the unicity property stated in Theorem 3.4. We retain the notations of Section 4 (before Theorem 4.3). In particular, we x a partition of k, and set = (0) for 2 Sr . We denote by W the subspace of Fq[Matn] whose decomposition as a Uq (gl n)-bimodule is W ' V V . Thus, W k is the homogeneous component of degree k of Fq [Matn]. For each , we choose a standard Young tableau of shape . We can now state Theorem 5.1 Let B; denote the following set of quantum bitabloids: ( )
B; = f(j0) j ; 0 have shape and Q(u) = Q(u0) = for some g : Then B; is a basis of W. The A-lattice L spanned by B; does not depend on . The projection B of B; in L =qL is also independent of , and (L; B ) is a crystal basis of W. Moreover for (j 0) 2 B; and ( j 0) 2 B; , we have
(j0) ( j 0) mod qL () P (u ) = P (u ) and P (u0 ) = P (u 0 ) : Finally, the crystal graph attached to (L; B ) is the bi-coloured graph given by il ( j0) () fiy(j0) = ( j0) () f^iu = u ; (j0) ?! ir (j 0) () fi(j0) = (j 0) () f^iu0 = u 0 : (j0) ?!
Proof: We imitate the proof of Theorem 4.3 and introduce the K (q)-linear map :
V
(11 )
V r V 1 V (c cr ) (d dr )
(1
1
)
(1
)
?! W ?! (c cr jd dr )
(1r )
1
1
1
Here ci; dj denote elements of the canonical bases of V i , V j , i.e. columns of size i , j , and (c cr jd dr ) is the quantum bitabloid formed on these columns. The map is in fact a homomorphism of Uq (gl n)-bimodules, and therefore sends the crystal lattice L; spanned by the tensors (c cr ) (d dr ) onto a crystal lattice L; in W. To describe precisely L; , it is enough to determine its submodule L; of highest weight vectors. We shall prove that L; = A (y jy ). To do this we have to introduce some notations. We write for short V := V 1
V r , and we consider a source vertex (1
1
)
(1
)
1
1
1
+
+
(1
(1
)
yk?r y . y = .. ... y1 yk 1
20
+1
)
of the crystal graph of V . This means that y yk is a Yamanouchi word on f1; : : : ; ng such that the columns of y are increasing from bottom to top. By de nition of a crystal basis, there exists a highest weight vector Ty in V such that Ty y mod qL; . Now, let 1
wk?r w . . w = . ... w1 wk 1
+1
be any monomial tensor of the same weight as y. Then (Ty w) = y;w (q) (y jy ) for some scalar y;w (q) 2 K (q). Indeed,
eyi (Ty w) = (eyi Ty w) = ((eiTy ) w) = 0 ; and on the other hand (Ty w) has weight (; ). Therefore (Ty w) is a scalar multiple of the highest weight vector (y jy ). But from the de nition of , we have (Ty w) =
X
u
u(q) (ujw) :
(33)
Here w is the tabloid made of the columns of w, u runs through the tabloids of the same shape and weight as w, and u(q) 2 A. Moreover, y (0) = 1 and u(0) = 0 for u 6= y. Expanding (33) on the basis
w = ftu1 w1 tuk wk j u uk has weight and wi = wj =) ui uj g 1
and comparing to the similar expansion of (y jy ), we obtain by checking the coecient of tw1 w1 twk wk that y;w (q) = w (q). Therefore, (Ty w) w y (y jy ) mod q(y jy ), and if y0 is another source vertex of ?(V ), we have (Ty Ty0 ) y y0 (y jy ) mod q(y jy ) :
(34)
This proves that L; = A (y jy ), as stated. It follows that L; does not in fact depend on , and we may write L; = L. We can now consider the map induced by from L; =qL; to L=qL (we still denote it by ). By (34), it maps to zero all the connected components of the crystal graph of V V with source vertex y y0 such that y 6= y0. This means precisely that (j0) 0 mod qL whenever Q(u) 6= Q(u0 ). On the other hand, to each corresponds a unique source vertex y = y( ) such that Q(uy ) = . The vertices w w0 of the connected components of ?(V V ) with origin y( ) y( ), , span a subspace of L; =qL; isomorphic under to L=qL. Therefore (L; B; mod qL) is a crystal basis of W, and this implies that B; mod qL is independent of . The other statements follow now easily from Theorem 3.8. 2 +
Proof of Theorem 1.1: Denote by L the crystal lattice of Fq [Matn ] whose submodule of highest weight vectors is equal to A (y jy ). It follows from the proof of Theorem 5.1 that the crystal lattice L of W is nothing but L \ W. We also see that L is spanned over A by the set of bitableaux ( j 0). Indeed, if we choose to be the standard
21
Young tableau whose rst column contains 1; 2; : : : ; 0 , whose second column contains 0 + 1; : : : ; 0 + 0 , and so on, we have 1
2
1
1
BT := f( j 0) j ; 0 are Young tableaux of shape g B;
id
and BT mod L is the part of B which labels the connected component of ?(W) corresponding to V V. Therefore, (L; tBT mod L) is a crystal basis of Fq[Matn] and L = A BT. On the other hand, taking = (k) in Theorem 5.1, we nd that the set of monomials
Bk = fti1j1 tik jk j Q(i ik ) = Q(j jk ) = for some g 1
1
gives rise to the same crystal basis under the guise (L; tk Bk mod L). Hence, applying again Theorem 3.4, it follows from the description of the crystal graph that mod qL if Q(i ik ) = Q(j jk ) (35) ti1j1 tik jk (P (i ik )0jP (j jk )) mod qL otherwise Finally, it is known that the coecients of the straightening of a bitabloid belong to K [q; q? ]. Since we have just proved that they also belong to A, we can replace in (35) the crystal lattice L by the K [q]-lattice L spanned by the quantum bitableaux. 2
1
1
1
1
1
References
[1] A. D. Berenstein, A. V. Zelevinsky, Canonical bases for the quantum group of type Ar and piecewise-linear combinatorics, Preprint Institut Gaspard Monge, 1994. [2] C. Burdik, R. C. King, T. A. Welsh, The explicit construction of irreducible representations of the quantum algebras Uq (sln ), preprint 1993. [3] V. Chari, A. Pressley, A guide to quantum groups, Cambridge Universty Press, 1994. [4] E. Date, M. Jimbo, T. Miwa, Representations of Uq (gln ) at q = 0 and the RobinsonSchensted correspondence, in Physics and mathematics of strings, Memorial volume of V. Knizhnik, L. Brink, D. Friedan, A.M. Polyakov eds., World Scienti c, 1990. [5] J. Desarmenien, An algorithm for the Rota straightening formula, Discrete Math. 30 (1980), 51-68. [6] J. Desarmenien, J.P.S. Kung, G.C. Rota, Invariant theory, Young bitableaux and combinatorics, Adv. Math. 27 (1978), 63-92. [7] V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254-258. [8] W. Fulton, A. Lascoux, A Pieri formula in the Grothendieck ring of a ag bundle, Duke Math. J., 76 (1994), 711-729. [9] J.A. Green, Classical invariants and the general linear group, in Representation theory of nite groups and nite dimensional algebras, G.O. Michler, C.M. Ringel eds., Progress in Mathematics 95, Birkhauser, 1991. [10] R.Q. Huang, J.J. Zhang, Standard basis theorem for quantum linear groups, Adv. Math. 102 (1993), 202-229.
22
[11] M. Jimbo, A q-di erence analogue of U (g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. [12] M. Jimbo, A q-analogue of U (gl(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252. [13] M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Commun. Math. Phys. 133 (1990), 249-260. [14] M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516. [15] M. Kashiwara, T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, 165 (1994), 295-345. [16] D.E. Knuth, Permutations, matrices and generalized Young tableaux, Paci c. J. Math. 34 (1970), 709-727. [17] A. Lascoux, B. Leclerc, J.Y. Thibon, Crystal graphs and q-analogues of weight multiplicities for root systems of type An , Lett. Math. Phys. to appear. [18] V. Lakshmibai, N. Yu. Reshetikhin, Quantum deformations of SLn =B and its Schubert varieties, in Special Functions, ICM-90 Satellite Conference Proceedings, M. Kashiwara, T. Miwa eds., Springer. [19] A. Lascoux, M. P. Schutzenberger, Le monode plaxique, in Noncommutative structures in algebra and geometric combinatorics A. de Luca Ed., Quaderni della Ricerca Scienti ca del C. N. R., Roma, 1981 [20] A. Lascoux, M.P. Schutzenberger, Keys and standard bases, in Invariant theory and tableaux, D. Stanton ed., Springer, 1990. [21] B. Leclerc, On identities satis ed by minors of a matrix, Adv. in Math., 100 (1993), 101-132. [22] M. Noumi, H. Yamada, K. Mimachi, Finite dimensional representations of the quantum group GLq (n; C ) and the zonal spherical functions on Uq (n ? 1)nUq (n), Japan. J. Math., 19 (1993), 31-80. [23] V. Reiner, M. Shimozono, Key polynomials and a agged Littlewood-Richardson rule, J. Comb. Theory A, 70 (1995), 107-143. [24] N.Y. Reshetikhin, L.A. Takhtadzhyan, L.D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J.,1 (1990), 193{225. [25] G. de B. Robinson, On the representation theory of the symmetric group, Amer. J. Math. 60 (1938), 745-760. [26] C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179-191. [27] M. P. Schutzenberger, La correspondance de Robinson, in Combinatoire et representation du groupe symetrique, Strasbourg 1976, D. Foata ed., Springer Lect. Notes in Math. 579. [28] E. Taft, J. Towber, Quantum deformation of ag schemes and Grassman schemes I { A q-deformation of the shape-algebra for GL(n), J. of Algebra, 142 (1991), 1{36.
23
Leclercy and Jean-Yves Thibonz
q-alg/9504004 7 Apr 1995
March 1995.
Dedicated to Dominique Foata
Abstract
We show that the quantum straightening algorithm for Young tableaux and Young bitableaux reduces in the crystal limit q 7! 0 to the Robinson-Schensted algorithm.
1 Introduction Let K be a eld of characteristic 0, and X = (xij ) i; jn a matrix of commutative indeterminates. The ring K [xij ] may be regarded as the algebra F [Mat n] of polynomial functions on the space of n n matrices over K . A linear basis of this algebra is given by the bitableaux of Desarmenien, Kung, Rota [6], which are de ned in the following way. Given two (semistandard) Young tableaux and 0 of the same shape, with columns c ; : : :; ck and c0 ; : : :; c0k , the Young bitableau ( j 0) is the product of the k minors of X whose row indices belong to ci and column indices to c0i, i = 1; : : : ; k. For example, 1
1
1
4 3 5 6 1 1 2
5 2 4 7 1 3 5
x x x := x x x xx xx xx xx : x x x More generally, we shall call tabloid a sequence of column-shaped Young tableaux, and we shall associate to each pair ; 0 of tabloids of the same shape a bitabloid ( j 0) de ned as the product of minors indexed by the columns of and 0.
11
12
31
32
41
42
15 35 45
13
53
14 54
25
65
27 67
Partially supported by PRC Math-Info and EEC grant n0 ERBCHRXCT930400 L.I.T.P., Universite Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France Institut Gaspard Monge, Universite de Marne-la-Vallee, 2 rue de la Butte-Verte, 93166 Noisy-leGrand cedex, France y z
1
There exists an algorithm due to Desarmenien [5] for expanding any polynomial in K [xij ] on the basis of bitableaux. This is the so-called straightening algorithm (for bitableaux). In particular, the monomials xi1j1 xikjk , which obviously form another linear basis of K [xij ], can be expressed in a unique way as linear combinations of bitableaux. Thus, the straightening of x x x reads 23
2 1 3 2 + 1 3
11
3 1 2 3 3 1 2 + 1 2
32
= 1 2 3
1 2 3
2 2 1 3 ? 1 3
3
3 1 2
? 1 2 2 1 3
+
3 2 1
3 2 1
On the other hand, the Robinson-Schensted correspondence [25, 26, 27, 16] associates to any word w on the alphabet of symbols f1; : : : ; ng a pair (P (w); Q(w)) of Young tableaux of the same shape. For example, the image of the word w = 2 1 4 3 5 1 2 under this correspondence is the pair 4 2 3 5 1 1 2 ,
6 2 4 7 1 3 5
Both the straightening algorithm and the Robinson-Schensted algorithm are strongly connected with the representation theory of GLn . Indeed, the straightening algorithm allows to compute the action of GLn in (polynomial) irreducible representations, while the Robinson-Schensted correspondence was devised by Robinson to obtain a proof of the Littlewood-Richardson rule for decomposing into irreducibles the tensor product of two irreducible representations. The problem that we want to investigate is whether there exists any relation between the straightening algorithm and the Schensted algorithm. It turns out that to answer this question, one has to replace the algebra F [Mat n ] by its quantum analogue Fq [Mat n] [24]. This is the associative algebra over K (q) generated by n letters tij ; i; j = 1; : : :; n subject to the relations 2
tik til tik tjk til tjk tik tjl ? tjl tik for 1 i < j n and 1 k < l n. element det q = det q T :=
q? til tik ; (1) ? q tjk tik ; (2) tjk til ; (3) ? (q ? q) til tjk ; (4) The quantum determinant of T = (tij ) is the
= = = =
X
1
1
1
(?q)?` w t w1 : : : tnwn ( )
w2Sn
2
1
of Fq [Matn ]. Here Sn is the symmetric group on f1; : : : ; ng and `(w) denotes the length of the permutation w. The quantum determinant of T belongs to the center of Fq [Matn ]. More generally, for I = (i ; : : : ; ik) and J = (j ; : : :; jk ) one de nes the quantum minor of the submatrix TIJ as 1
1
det q TIJ :=
X
(?q)?` w ti1jw1 : : : tikjwk ; ( )
w2Sk
and the quantum bitableau ( j 0) (resp. quantum bitabloid ( j 0)) as the product of the quantum minors indexed by the columns of the Young tableaux and 0 (resp. of the tabloids and 0). As proved by Huang, Zhang [10], the set of quantum bitableaux is again a linear basis of Fq [Mat n]. For example, the expansion of the monomial t t t on this basis is 23 11 32
2 1 3
2 + (1 ? q + q ) 1 3 2
= q
3 1 2
3 1 2
4
3
1 2 3
1 2 3
3 +q 1 2
2 1 3
4
+q
3 2 1
5
3 ?q 1 2
3 1 2
2 1 3
2 1 3
3
? q
5
3 2 1
In the present example the coecients of the expansion are polynomials in q, and only one of them has a nonzero constant term. In other words, denote by B the linear basis of quantum bitableaux in Fq [Mat n], and let L be the lattice generated over K [q] by the elements of B. Then 2 3 1 2 1 3
2 1 3
3 1 2
mod qL
This is an illustration of the following Theorem 1.1 Let w = i ik and u = j jk be two words on f1; : : : ; ng, and denote by (P (w); Q(w)), (P (u); Q(u)) their images under the Robinson-Schensted correspondence. Then, qL if Q(w) = Q(u) : ti1j1 tik jk (P (w)0jP (u)) mod mod qL otherwise 1
1
3
As a corollary we obtain an unexpected characterization of the plactic congruence on words [16, 19], de ned by
w u () P (w) = P (u) :
Corollary 1.2 With the same notations as above, w u () ti1i1 tikik tj1j1 tjk jk mod qL : At this point, it is important to recall that the existence of a connection between the Robinson-Schensted correspondence and the representation theory of the quantized enveloping algebra Uq (gl n ) at q = 0 was rst discovered by Date, Jimbo, Miwa [4]. The results presented here are in fact of the same kind as those of [4], namely, it is shown in [4] that if V denotes the basic representation of Uq (gl n ), the transition matrix in V k from the basis of monomial tensors to the Gelfand-Zetlin basis specializes when q = 0 to a permutation matrix given by the Robinson-Schensted map. Similarly, Theorem 1.1 states that the transition matrix in the Uq (gl n )-module Fq [Mat n] from the basis of monomials (1)
(1)
B = fti1j1 tik jk j Q(i ik) = Q(j jk ) = for some g to the basis of quantum bitableaux is equal at q = 0 to a permutation matrix also computed from the Robinson-Schensted algorithm. Here, denotes for each partition a xed standard Young tableau of shape . The work of Date, Jimbo, Miwa, provided the starting point from which Kashiwara developed his theory of crystal bases for quantized enveloping algebras [13, 14]. We shall use crystal bases as the main tool for proving Theorem 1.1. The paper is organized as follows. In Section 2, we collect the necessary material about Uq (gl n), Uq (sl n) and their representation theory. In Section 3, we review the de nition and basic properties of Kashiwara's crystal bases at q = 0. In Section 4, we formulate and prove a version of Theorem 1.1 for the case of (single) tableaux, that is, we work in the subring of Fq[Mat n ] generated by the quantum minors taken on the initial rows of T . Finally in Section 5 we prove Theorem 1.1, as well as a slightly more general statement. 1
2
Uq (gl n)
1
and Uq(sl n)
A general reference for this Section and the following one is the excellent exposition [3]. We rst recall the de nition of the quantized enveloping algebras Uq (gl n ) [12] and Uq (sl n) [11, 7]. Uq (gl n) is the associative algebra over K (q) generated by the 4n ? 2 symbols ei; fi; i = 1; : : :; n ? 1 and qi ; q?i ; i = 1; : : :; n, subject to the relations
qi q?i = q?i qi = 1 ; [qi ; qj ] = 0 ; qej for i = j qi ej q?i = q? ej for i = j + 1 ; ej otherwise 8 >
:
4
(5) (6)
q? fj for i = j (7) qi fj q?i = qfj for i = j + 1 ; fj otherwise i ?i+1 ?i i+1 [ei; fj ] = ij q q q ??qq? q ; (8) [ei; ej ] = [fi; fj ] = 0 for ji ? j j > 1 ; (9) ej ei ? (q + q? )eiej ei + ei ej = fj fi ? (q + q? )fifj fi + fi fj = 0 for ji ? j j = 1 : (10) The subalgebra of Uq (gl n) generated by ei; fi, and qhi = qi q?i+1 ; q?hi = q?i qi+1 ; i = 1; : : : ; n ? 1; (11) is denoted by Uq (sl n). The representation theories of Uq (gl n ) and Uq (sl n) are closely parallel to those of their classical counterparts U (gl n) and U (sl n ). Let M be a Uq (gl n)-module and = ( ; : : : ; n) be a n-tuple of nonnegative integers. The subspace M = fv 2 M j qi v = qi v ; i = 1; : : : ; ng is called a weight space and its elements are called weight vectors (of weight ). Relations (6) (7) show that eiM M+ ; fiM M? ; where = ( ; : : :; i + 1; i ? 1; : : : ; n ) and ? = ( ; : : :; i ? 1; i + 1; : : : ; n). Thus, ordering the weights in the usual way by setting 8 >
:
1
2
1
2
2
1
2
1
+
1
+1
()
1
k
X
i=1
i
k
X
i=1
i ;
+1
k = 1; : : : ; n;
we see that the ei's act as raising operators and the fi's as lowering operators. A weight vector is said to be a highest weight vector if it is annihilated by the ei's. M is called a highest weight module if it contains a highest weight vector v such that M = Uq (gl n ) v. If v is of weight , it follows that dim M = 1 and M = M . One then shows that there exists for each partition of length n a unique highest weight nite-dimensional irreducible Uq (gl n)-module V, with highest weight .
Example 2.1 The basic representation V = V of Uq (gl n ) is the n-dimensional vector space over K (q) with basis fvi; 1 i ng, on which the action of Uq (gl n ) is as follows: (1)
qi vj = qij vj ; eivj = i j vi; fi vj = ij vi : +1
+1
Example 2.2 More generally, the Uq (gl n )-module V k is a nk -dimensional vector space with basis fvcg labelled by the subsets c of f1; : : : ; ng with k elements (i.e. by the Young tableaux of shape (1k ) over f1; : : : ; ng). The action of Uq (gl n) on this basis is given by qi vc = vc if i 62 c ;
(1 )
qvc otherwise 5
i + 1 62 c or i 2 c eivc = v0 otherwise; ifwhere d = (c n fi + 1g) [ fig ; d i + 1 2 c or i 62 c fivc = v0 otherwise; ifwhere d = (c n fig) [ fi + 1g : d We see that the action of the lowering operators fi does not depend on q, and can be recorded on a colored graph whose vertices are the column-shaped Young tableaux c and whose arrows are given by: i c ?! d () fivc = vd : Thus for k = 2; n = 4, one has the following graph: 2 2 3 3 4 1 1 1
1 3 2
1 3
4 2 2 4 3
This is one of the simplest examples of crystal graphs (cf. Section 3). In order to construct more interesting Uq (gl n )-modules, we use the tensor product operation. Given two Uq (gl n )-modules M; N , we can de ne a structure of Uq (gl n )module on M N by putting qi (u v) = qi u qi v; (12) ? h i ei(u v) = eiu v + q u eiv; (13) h i fi(u v) = fiu q v + u fiv: (14) Indeed, the formulas qi = qi qi ; ei = ei 1 + q?hi ei; fi = fi qhi + 1 fi; de ne a comultiplication on Uq (gl n). One shows that the decomposition into irreducible components of the tensor product of two irreducible Uq (gl n )-modules is given by V V ' c V ; (15) M
where the c are the classical Littlewood-Richardson numbers. In particular, it follows that V k ' f V ; (16) M
`k
where f denotes the number of standard Young tableaux of shape . 6
Example 2.3 The n -dimensional Uq (gl n )-module V decomposes into the q-symmetric 2
2
square V and the q-alternating square V by the following diagram: (2)
;
(1 1)
. For n = 2 this decomposition is described
f1 f1 f1 0 e1? v v ?! v v + qv v ?! (q + q? ) v v ?! 0 'V 1
1
1
2
2
1
1
2
f1 0 e1? v v ? qv v ?! 0 'V 2
1
1
2
2
(2)
;
(1 1)
The algebra Fq [Mat n] de ned in Section 1 is also endowed with a natural structure of Uq (gl n)-module via the action de ned by
qi tkl = qil tkl; ei tkl = i l tk l? ; fi tkl = il tk l ; (17) and the Leibniz formulas (18) qi (PQ) = (qi P ) : (qi Q); ? h i ei(PQ) = (eiP ) : Q + (q P ) : (eiQ); (19) h (20) fi(PQ) = (fiP ) : (q i Q) + P : (fiQ); for P; Q in Fq [Mat n]. This provides a very convenient realization of the irreducible modules V as natural subspaces of Fq [Mat n]. To describe it, we introduce some notations. We shall write y for the unique Young tableau of shape and weight . This is the socalled Yamanouchi tableau of shape . Let be any Young tableau of shape . The quantum bitableau (y j ) will be simply denoted by ( ) and will be called a quantum tableau. This is a product of quantum minors taken on the rst rows of the matrix T . Quantum tabloids are de ned similarly. Finally, denote by T the subspace of Fq [Mat n ] spanned by quantum tableaux ( ) of shape . Then one can show [18, 22] the following q-analogue of a classical result of Deruyts (see [9]). Theorem 2.4 The subspace T is invariant under the action of Uq (gl n) on Fq [Mat n ], and is isomorphic as a Uq (gl n)-module to the simple module V . The action of Uq (gl n ) on T is computed by means of the q-straightening formula. Namely, for column-shaped quantum tableaux one checks easily that the action coincides with the one previously described in Example 2.2. For general quantum tableaux we use Leibniz formulas (18) (19) (20), and when necessary we use the q-straightening algorithm (cf. Section 4) for converting the quantum tabloids of the right-hand side into a linear combination of quantum tableaux. Example 2.5 We choose n = 3 and = (2; 1). T ; is 8-dimensional and one has for instance 3 3 3 3 2 f 1 1 = 1 2 + q 2 1 = (1 + q ) 1 2 ?q 1 3 +1
1
+1
(2 1)
2
1
3
In this example, the quantum tableaux ( ) have been written for short (without brackets). This small abuse of notation will be used freely in the sequel. 7
This realization of V is, up to some minor changes of convention, the same as the one described in [2] via the q-Young symmetrizers of the Hecke algebra of type A. We denote by Fq[GLn =B ] the subspace of Fq [Mat n] spanned by the quantum tableaux. It follows from the q-straightening formula that this is in fact a subalgebra of Fq[Mat n ]. It coincides for q = 1 with the ring of polynomial functions on the ag variety GLn =B , hence the notation. The quantum deformation Fq [GLn =B ] has been studied by Lakshmibai, Reshetikhin [18] and Taft, Towber [28]. As a Uq (gl n )-module it decomposes into:
Fq [GLn =B ] '
M
`()n
V :
(21)
Returning to Fq [Mat n ] and its linear basis formed by quantum bitableaux, we note that the action of Uq (gl n) de ned by (17) involves only the column indices of the variables tij . The de ning relations (1) (2) (3) (4) being invariant under transposition of the matrix T , we see that we have another action of Uq (gl n ) given by (qi )ytkl = qik tkl; eyi tkl = i
fiytkl = ik tk l; (22) (where the symbol y has been added to distinguish this action from the previous one), and the Leibniz formulas (18) (19) (20). These two actions obviously commute with each other, so that Fq [Mat n] is now endowed with the structure of a (left) bimodule over Uq (gl n). The quantum version of the Peter-Weyl theorem provides the decomposition [22] Fq [Mat n ] ' V V : (23) +1
k tk?1 l ;
+1
M
`()n
Here, the irreducible bimodule V V is generated by applying all possible products of lowering operators fiy; fj to the highest weight vector (yjy). We end this Section by noting that every Uq (gl n)-module M can be regarded by restriction as a Uq (sl n )-module (that we still denote by M ). In particular, the V are also irreducible under Uq (sl n ). However we point out that, as Uq (sl n)-modules,
V ' V () i ? i = i ? i ; i = 1; : : : ; n ? 1 : +1
+1
Example 2.6 The Uq (sl )-modules V l will be very important in the sequel and we describe them precisely. For l 0, V l is a (l + 1)-dimensional vector space over K (q) with basis fuk ; 0 k lg, on which the action of Uq (sl ) is as follows: qh1 uk = ql? k uk ; e uk = [l ? k + 1] uk? ; f uk = [k + 1] uk : In these formulas [m] denotes the q-integer (qm ? q?m)=(q ? q? ), and we understand u? = ul = 0. Setting [m]! = [m][m ? 1] [1] and f m = f m =[m]!, we see that the basis fuk g is characterized by uk = f k u . Also, we note that the weight spaces being one-dimensional, there is up to normalization a unique basis of V l whose elements are weight vectors. The basis fuk g may therefore be regarded as canonical. This will provide 2
()
()
2
2
1
1
1
+1
1
1
+1
( ) 1
( 1
)
1
0
()
the starting point for de ning the crystal basis of a Uq (gl n )-module. 8
3 Crystal bases
It follows from relations (6) (7) (8) (11) that for any i = 1; : : :; n ? 1, the subalgebra Ui generated by ei; fi; qhi ; q?hi is isomorphic to Uq (sl ). Hence a Uq (gl n )-module M can be regarded by restriction to Ui as a Uq (sl )-module. We shall assume from now on that the weight spaces M are nite-dimensional, that M = M , and that for any i, M decomposes into a direct sum of nite-dimensional Ui-modules. Such modules M are said to be integrable. It follows from the representation theory of Uq (sl ) that for any i, the integrable module M is a direct sum of irreducible Ui-modules V l . Example 3.1 Let M denote the Uq (gl )-module V ; in the realization given by Theorem 2.4. As a U -module, M decomposes into 4 irreducible components, as shown by the following diagram: 2
2
2
()
3
(2 1)
1
0
e
1
0
e
1
0
e
1
0
e
1
2 1 1
f
3 1 1
f
2 1 3
f
3 1 3
f
2 1 2
1
f
0
1
3 (1 + q ) 1 2 ? q
1
2
3
2 1 3
f =[2] 1
3 2 2
f
0
1
0
1
3 2 3
1
f
0
1
On the other hand, as a U -module, M decomposes into: 2
0
e
2 1 1
f
0
e
2 1 2
f
0
e
3 1 2 ?q
0
e
3 2 2
2
2
2
2
3 1 1
2
2
2
2 1 3 +q
2
2 1 3
f
f
f
0 3 1 2
0
2
3 2 3
f 9
2
0
f =[2] 2
3 1 3
f
2
0
We observe that the U -decomposition leads to the basis 1
2 B =( 1 3 + q 2
for the 2-dimensional weight space M 2 B =( 1 3 + q 2
3 1 2 ;;
(1 1 1)
3 ; 1 2 ?q
2 1 3 )
, while the U -decomposition leads to 2
3 1 2
3 ; 1 2 ?q
2 1 3 )
These two bases are di erent and therefore one cannot nd a basis B of weight vectors in M compatible with both decompositions. However, as noted by Kashiwara, `at q = 0' the bases B and B coincide. The next de nitions will allow us to state this in a more formal way. Consider the simple Uq (sl )-module V l with basis fuk g (cf. Example 2.6). Kashiwara [13, 14] introduces the endomorphisms e~; f~ of V l de ned by e~ uk = uk? ; f~ uk = uk ; k = 0; : : : ; l; 1
2
2
()
()
1
+1
where u? = ul = 0. More generally, if M is a direct sum of modules V , that is, if there V l , one de nesl endomorphisms exists an isomorphism of Uq (sl )-modules : M ?! l ~ e~; f of M by means of in the obvious way, and one checks easily that they do not depend on the choice of . In particular, if M is an integrable Uq (gl n )-module, regarding M as a Ui -module we de ne operators e~i; f~i on M for i = 1; : : : ; n ? 1. Example 3.2 We keep the notations of Example 3.1. We have 2 3 3 f~ 1 3 +q 1 2 1 3 1
+1
()
2
()
2
3 2 1 2 ?q 1 3
f~
0
2
Hence, 2 1 3
1 31 3 1+q 3 q 31 3 f~ 1 2 1+q Since we want to let q tend to 0, we introduce the subring A of K (q) consisting of rational functions without pole at q = 0. A crystal lattice of M is a free A-module L such that M = K (q) A L; L = L where L = L \ M, and e~iL L ; f~iL L ; i = 1; : : : ; n ? 1 : (24)
f~
2
2
2
2
10
In other words, L spans M over K (q), L is compatible with the weight space decomposition of M and is stable under the operators e~i and f~i. It follows that e~i; f~i induce endomorphisms of the K -vector space L=qL that we shall still denote by e~i; f~i. Now Kashiwara de nes a crystal basis of M (at q = 0) to be a pair (L; B ) where L is a crystal lattice in M and B is a basis of L=qL such that B = tB where B = B \ (L =qL), and e~iB B t f0g ; f~iB B t f0g ; i = 1; : : : ; n ? 1 ; (25)
e~iv = u () f~iu = v; u; v 2 B; i = 1; : : : ; n ? 1 :
(26)
Example 3.3 We continue Examples 3.1 and 3.2. Denote by B the basis of quantum tableaux in M , and let L be the A-lattice in M spanned by the elements of B. Let B be the projection of B in L=qL. Then, Example 3.1 shows that (L; B ) is a crystal basis of M.
Kashiwara has proven the following existence and uniqueness result for crystal bases [13, 14]. Theorem 3.4 Any integrable Uq (gl n )-module M has a crystal basis (L; B ). Moreover, if (L0 ; B 0) is another crystal basis of M , then there exists a Uq (gl n )-automorphism of M sending L on L0 and inducing an isomorphism of vector spaces from L=qL to L0 =qL0 which sends B on B 0 . In particular, if M = V is irreducible, its crystal basis (L(); B ()) is unique up to an overall scalar multiple. It is given by A f~i1 f~i2 f~ir u; (27) L() = X
i1 ;i2 ;:::;ir n?1
1
B () = ff~i1 f~i2 f~ir u mod qL() j 1 i ; : : : ; ir ngnf0g; (28) where u is a highest weight vector of V . It follows that one can associate to each integrable Uq (gl n )-module M a well-de ned colored graph ?(M ) whose vertices are labelled by the elements of B and whose edges describe the action of the operators f~i : 1
i u ?! v () f~iu = v : ?(M ) is called the crystal graph of M . Example 3.5 The crystal graph of the Uq (gl )-module V ; is readily deduced from Examples 3.1, 3.2 and 3.3. It is shown in Figure 1. It is clear from this de nition that the crystal graph ?(M ) of the direct sum M = M M of two Uq (gl n )-modules is the disjoint union of ?(M ) and ?(M ). It follows from the complete reducibility of M that the connected components of ?(M ) are the crystal graphs of the irreducible components of M . Note also that if one restricts the crystal graph ?(M ) to its edges of colour i, one obtains a decomposition of this graph into strings of colour i corresponding to the Ui-decomposition of M . For a vertex v of 3
1
2
(2 1)
1
11
2
2 11
3 11
2
3 12
1
1
1
2 12
2
2 13
3 22
2
2
3 13
3 23
1
Figure 1: Crystal graph of the Uq (gl )-module V 3
;
(2 1)
?(M ), we shall denote by i(v) (resp. i(v)) the distance of v to the origin (resp. end) of its string of colour i, that is, i(v) = maxfk j e~ki v 6= 0g ; i(v) = maxfk j f~ik v 6= 0g : The integers i(v), i(v) give information on the geometry of the graph ?(M ) around the vertex v. They seem to be very signi cant from a representation theoretical point of view. Indeed, both the Littlewood-Richardson multiplicities c and the q-weight multiplicities K (q) can be computed in a simple way from the integers i(v), i(v) attached to the crystal graph ?(V ) [1, 17]. One of the nicest properties of crystal bases is that they behave well under the tensor product operation. We shall rst consider an example, from which Kashiwara deduces by induction the general description of the crystal basis of a tensor product [13].
Example 3.6 We slightly modify the notations of Example 2.6 and write fukl ; k = 0; : : : ; lg for the canonical basis of the Uq (sl )-module V l . We recall that for convenience we set u?l = ul l = 0. A rst basis of the tensor product V V l is fuj ukl ; j = 0; 1; k = 0; : : :; lg. We de ne wk = u ukl + ql?k u ukl? ; k = 0; : : :; l + 1 ; (29) l l zk = ql?k? [l ? k] u uk ? ql [k + 1] u uk ; k = 0; : : : ; l ? 1 : (30) ()
2
() 1
()
() +1
(1)
(1) 0
()
+1
(1) 1
1
(1) 1
()
(1)
()
1
()
(1) 0
() +1
It is straightforward to check that e w = e z = 0; f wl = f zl? = 0, and 1
0
1 0
1
+1
1
f wk = [k + 1] wk ; f zk = [k + 1] zk : 1
+1
1
12
+1
1
()
Hence fwk g and fzk g span submodules of V V l isomorphic to V l and V l? respectively, and are the canonical bases of these irreducible representations of Uq (sl ). Therefore, the A-lattice L spanned by fwk g, fzk g is a crystal lattice and if we de ne B = fwk mod qLgtfzk mod qLg, then (B; L) is a crystal basis of V V l . On the other hand, equations (29) (30) show that L coincides with the tensor product L(1) L(l) of the crystal lattices of V and V l , and that (1)
()
( +1)
(
1)
2
(1)
(1)
()
()
wk u ukl mod qL ; k = 0; : : : ; l ; wl u ul l mod qL : (1) 0
()
(1) 1
+1
()
zk u ukl mod qL ; k = 0; : : : ; l ? 1 : Thus, (L; B ) coincides with (L(1) L(l); B (1) B (l)), where we have denoted by B (1)
B (l) the basis fuj ukl ; mod qL ; j = 0; 1; k = 0; : : : ; lg. Finally, the action of f~ on B (1) B (l) is described by the crystal graph: (1) 1
(1)
()
()
1
u u l ! u u l ! ! u ul?l ! u ul l (1) 0
() 0
(1) 0
() 1
(1) 0
() 1
(1) 0
()
# l u ul
u u ! u u ! ! u ul? More generally, we have the following property [13]. Theorem 3.7 Let (L ; B ) and (L ; B ) be crystal bases of integrable Uq (gl n )-modules M and M . Let B B denote the basis fu v; u 2 B ; v 2 B g of (L =qL ) (L =qL ) (which is isomorphic to (L L )=q(L L )). Then, (L L ; B B ) is a crystal basis of M M , the action of e~i, f~i on B B being given by ~ i(v) ; f~i(u v) = uf~ u
fivv ifif i((uu)) < (31) i(v) i i > i(v) : e~i(u v) = eu~iu
e~ vv ifif i((uu)) (32) (v) l
(1) 1
() 0
1
2
1
l
(1) 1
() 1
1
2
l
() 1
(1) 1
()
2
1
2
1
1
1
(1) 1
2
1
2
2
2
1
1
1
2
1
1
2
2
2
2
i
i
i
Theorem 3.7 enables one to describe the crystal graph of V m for any m, and to deduce from that the description of the crystal graph ? of the simple Uq (gl n )-module V. For convenience, we shall identify the tensor algebra T (V ) with the free associative algebra K (q)hAi over the alphabet A = f1; : : : ; ng via the isomorphism vi 7! i, where fvig is the canonical basis of V de ned in Example 2.1. Accordingly, the crystal lattice L spanned by the monomials in the vi is identi ed with AhAi and the K -vector space L=qL with K hAi. De ne linear operators e^i, f^i on K hAi in the following way. Consider rst the case of a two-letter alphabet A = fi; i+1g. Let w = x xm be a word on A. Delete every factor ((i+1) i) of w. The remaining letters constitute a subword w of w. Then, delete again every factor ((i+1) i) of w . There remains a subword w . Continue this procedure until it stops, leaving a word wk = xj1 xjr+s of the form wk = ir(i+1)s. The image of wk under e^i; f^i is the word yj1 yjr+s given by (1)
(1)
(1)
1
1
1
e^i(ir(i+1)s ) =
ir (i+1)s? (s 1) ; 0 (s = 0) +1
2
r? s f^i(ir(i+1)s ) = i (i+1) 0
1
13
1
+1
(r 1) : (r = 0)
The image of the initial word w is then w0 = y ym, where yk = xk for k 62 fj ; : : :; jr s g. For instance, if w = (2 1) 1 1 2 (2 1) 1 1 1 2 ; we shall have w = : : 1 1 (2 : : 1) 1 1 2 ; w = : : 1 1 : : : : 1 1 2: Thus, e^ (w) = 2 1 1 1 2 2 1 1 1 1 1 ; f^ (w) = 2 1 1 1 2 2 1 1 1 2 2 ; where the letters printed in bold type are those of the image of the subword w . Finally, in the general case, the action of the operators e^i; f^i on w is de ned by the previous rules applied to the subword consisting of the letters i; i+1, the remaining letters being unchanged. The operators e^i, f^i have been considered in [20], where they were used as building blocks for de ning noncommutative analogues of Demazure symmetrization operators. It follows from Theorem 3.7 that they coincide in the above identi cation with the endomorphisms e~i, f~i on the K -space L=qL [15]. It is straightforward to deduce from the de nition of e^i, f^i the following compatibility properties with the Robinson-Schensted correspondence: (a) for any word w on A such that e^i w 6= 0, we have Q(^ei w) = Q(w), (b) for any pair of words w, u such that P (w) = P (u) and e^iw 6= 0, we have P (^eiw) = P (^eiu). In other words, the operators e^i, f^i do not change the insertion tableau and are compatible with the plactic equivalence. Moreover, the words y such that e^iy = 0 for any i are characterized by the Yamanouchi property: each right factor of y contains at least as many letters i than i +1, and this for any i. It is well-known that for any standard Young tableau there exists a unique Yamanouchi word y such that Q(y) = . This yields the following cristallization of (16). Theorem 3.8 The crystal graph of the Uq (gl n)-module V m is the colored graph whose vertices are the words of length m over A, and whose edges are given by: i w ?! u () f^iw = u : The connected components ? of ?(V m ) are parametrized by the set of Young tableaux of weight (1m ). The vertices of ? are those words w which satisfy Q(w) = . Moreover, if denotes the shape of , then ? is isomorphic to the crystal graph ?(V). Replacing the vertices w of ? by their associated Young tableaux P (w), we obtain a labelling of ?(V) by the set of Young tableaux of shape , as shown in Example 3.5. It follows from property (b) above that this labelling does not depend on the particular choice of among the standard Young tableaux of shape . We end this section by showing a less trivial example of crystal graph, which is computed easily using the previous description of f^i. Example 3.9 The crystal graph of the Uq (gl )-module V ; is shown in Figure 2. 1
1
+
1
2
1
1
2
(1)
(1)
4
14
(2 2)
44 11
1
3
44 12
1
3 24 11
1
2
3 22 11
2
23 11
34 1 11
24 12
2 3
1
2
3
23 12
33 11 2
3 1
33 12
2 44 22
44 13 2
1
44 23
3
2
44 33
3 34 3 12 1 34 13 1 24 2 13 34 2 34 22 23 3 1
33 22
Figure 2: Crystal graph of the Uq (gl )-module V 4
4 Fq[GLn=B ]
;
(2 2)
The subalgebra Fq[GLn =B ] of Fq[Matn] generated by quantum column-shaped tableaux can also be de ned by generators and relations [28, 18]. The generators are denoted by columns ik ... 1 k n; 1 i ; : : : ; ik n; i i and their products are written by juxtaposition. The relations are 1
2 1
15
(R ) 1
if ir = is for some indices r; s, then
ik ... = 0; i i 2 1
(R ) 2
for w 2 Sk and i < i < < ik , we have 1
2
iwk ... = (?q)?` w iw2 iw1
( )
(R ) 3
ik ... ; i i 2 1
for k l and j < j < < jl, we have 1
2
jwl ...
jl ...
jk j. k = .. j +1
i. k .. i
(?q)`(w)
X
w
1
1
1
jwk+1 i.k .. i
jw. k .. jw1
where the sum runs through all w 2 Sl such that w < < wk and wk < < wl, (R ) for k + s l ? 1 and j < j < < jl, we have 1
4
1
+1
2
(?q)` w
X
w
( )
jwl ... jwk+2 jwk+1 ik ? ... i 1
1
rs ... r jwk = 0 jwk?1 ... jw1 1
where the sum runs through all permutations w 2 Sl such that w < < wk and wk < < wl. Relations (R ), (R ) express the q-alternating property of quantum minors, while relations (R ), (R ) are the natural q-analogues of Sylvester's and Garnir's identities respectively (cf. [21]). Using (R ), (R ), (R ) one can express every element of Fq[GLn =B ] as a linear 1
+1
1
3
2
4
1
2
3
16
combination of products of columns, the size of which weakly decreases from left to right. After that, following the same strategy as in the classical straightening algorithm, one can by means of (R ) express all such products as linear combinations of quantum tableaux. Note that it follows from (R ), (R ) that all the quantum tableaux ( ) appearing in the straightening of a given tabloid () have the same shape obtained by reordering the sizes of the columns of , which means that () falls in the irreducible component V of Fq [GLn =B ]. Example 4.1 We apply the q-straightening algorithm to 4
3
4
6 () = 51 32 It follows from (R ), (R ), (R ) that 1
2
3
6 6 5 5 5 3 = 5 3 + 3 6 ?q 2 6 1 2 1 2 1 2 1 3 Using (R ) for straightening the rst term of the right-hand side, we obtain 4
6 5 3 =(1 ? q 1 2
2
5 5 3 6 ) +(q ? q) 2 6 1 2 1 3
6 +q 3 5 ?q 1 2
3
2
6 2 5 1 3
3 ?q 2 6 1 5 4
Letting q tend to 0 in this expansion, we get an illustration of the next Theorem, which can be regarded as a version of Theorem 1.1 for Fq [GLn =B ].
Theorem 4.2 Let B denote the linear basis of Fq [GLn=B ] consisting of quantum tableaux, and let L be the K [q]-lattice generated by B. Let be a partition, and () a quantum tabloid
in V. Denote by w the word obtained by reading the columns of from top to bottom and left to right. Then, qL if P (w ) has shape : () (P (0w)) mod mod qL otherwise
The proof of Theorem 4.2 is derived from the following result which describes several interesting bases of V, and states that each of them gives rise to the same crystal basis at q = 0. We rst introduce some notations. Let be a partition, 0 = (0 ; : : : ; 0r ) its conjugate, and (0) = (01 ; : : :; 0r ) the composition obtained by permuting the parts of 0 using the permutation 2 Sr . A tabloid is said to have shape (0) if it is a sequence of column-shaped Young tableaux (c ; : : : ; cr), the size of the column ci being equal to 0i , i = 1; : : :; r. The column reading of is the word u obtained by reading the columns of from top to bottom and left to right. 1
1
17
Theorem 4.3 For any 2 Sr, the set of quantum tabloids B = f() j has shape (0) and P (u ) has shape g is a linear basis of V. The A-lattice L spanned by B is independent of as well as the projection B of B in L=qL, and (L; B ) is a crystal basis of V. For () 2 B and ( ) 2 B , we have () ( ) mod qL () P (u ) = P (u ) : Finally, if () is such that P (u ) has not shape , then () 0 mod qL. Proof: We x 2 Sr and we write = (1 ; : : :; r ) := (0). Recall from Example 2.2 that the q-analogue V(1k ) of the k th exterior power of the basic representation has a natural basis labelled by column-shaped Young tableaux. We introduce the K (q)-linear map ':
V
1
(1 )
V
2
(1 )
V
?! V
(1r )
b. d... .. ..
a c.
f ... ?! e
b. .. a
d. .. ... f... e c
Comparing (12) (13) (14) to (18) (19) (20) we see immediately that ' commutes with the action of Uq (gl n ) and hence is in fact a morphism of Uq (gl n )-modules. The image V is irreducible and by (15) it appears with multiplicity one in V 1 V r . Therefore, the restriction of ' to the irreducible component of V 1 V r with highest weight vector the tensor product y := (y 1 ) (y r ) of all highest weight vectors in V 1 ; : : :; V r is an isomorphism, and the kernel of ' is the sum of all components V ; 6= appearing in V 1 V r . We now turn to the crystal bases of these various Uq (gl n)-modules. It follows from Example 2.2 and Theorem 3.4 that, if we choose u k = y k , L(1k ) is the A-lattice spanned by the basis of column-shaped quantum tableaux, while B (1k ) is the projection of this basis in L(1k )=qL(1k ). Using Theorem 3.7 we deduce that a crystal basis of V 1 V r can be constructed by tensoring the crystal bases of the factors. Denote this basis by (L; B ). Then '(L ) is a crystal lattice in V. According to Theorem 3.4 such a lattice is unique up to an overall scalar multiple which can be determined by considering the highest weight space. Here we have (L) = A y and ' (y) = (y). Therefore, '(L ) is the crystal lattice L of V speci ed by L = A (y), and it does not depend on . Now ' induces an isomorphism from the K -subspace of L=qL spanned by the subset of B consisting of those b in the connected component of the crystal graph with source y mod qL, to the K -space L=qL. It is easy to check from Theorem 3.7 and the explicit description of the operators e^i, f^i which follows it that this connected component is labelled by the elements b = c cr mod L such that the word (1
(1
(1
(1
)
)
(1
)
(1
(1
)
1
18
(1 )
)
(1
(1
)
(1 )
)
)
)
)
(1
(1
(1
)
)
ub obtained by reading c from top to bottom, then c from top to bottom, and so on, satis es: P (ub) has shape . Hence its image is B mod qL. This proves that B is a basis of V and that L = b2B A b. Moreover, (L; B mod qL) is a crystal basis, which proves by unicity that B mod qL does not depend on . On the other hand, the elements b of B which belong to the other connected components are sent to 0 in L=qL. This means that the tabloids () such that P (u ) has not shape belong to qL. Finally, if is another permutation and a tabloid of shape (0), the fact that () ( ) mod qL means exactly that the words u and u label the same vertex in the two copies of ?(V ) to which they belong in ?(V k ), that is, by Theorem 3.8, P (u ) = P (u ). 1
2
P
(1)
2
Proof of Theorem 4.2: Let () be a quantum tabloid in V. The q-straightening algorithm shows that () is a K [q; q?1]-linear combination of quantum tableaux. On the other hand, Theorem 4.3 shows that () belongs to the A-lattice L spanned by the set B = Bid of quantum tableaux. Therefore, () is in fact an element of the K [q]-lattice L spanned by B. The other statements of Theorem 4.2 follow immediately from Theorem 4.3. 2
We point out that the quantum tabloids () 2 B are easily computed using Schutzenberger's jeu de taquin [27]. For example, the following graph shows the 's giving rise to quantum tabloids () which are congruent to the quantum tableau ( ) modulo qL:
1
3 4 1 3 2 5
2
3 1 4 5 3 2
1
4 = 3 3 1 2 5
3 4 5 1 3 2 2
4 3 1 3 5 2
1
3 4 3 1 5 2
2
The edges labelled i connect tabloids ; obtained one from the other by permuting the column lengths of the i th and (i+1) th columns by means of jeu de taquin. The column readings u of these tabloids are distinguished words of the plactic class of u , called frank words [20]. The combinatorics of frank words occurs in several interesting problems (see [20, 23, 8]).
19
5 Proof of Theorem 1.1 The proof of Theorem 1.1 follows the same lines as that of Theorem 4.2, namely we rst describe several realizations of the crystal basis of the bimodule Fq [Matn] and then deduce Theorem 1.1 by comparing them, using the unicity property stated in Theorem 3.4. We retain the notations of Section 4 (before Theorem 4.3). In particular, we x a partition of k, and set = (0) for 2 Sr . We denote by W the subspace of Fq[Matn] whose decomposition as a Uq (gl n)-bimodule is W ' V V . Thus, W k is the homogeneous component of degree k of Fq [Matn]. For each , we choose a standard Young tableau of shape . We can now state Theorem 5.1 Let B; denote the following set of quantum bitabloids: ( )
B; = f(j0) j ; 0 have shape and Q(u) = Q(u0) = for some g : Then B; is a basis of W. The A-lattice L spanned by B; does not depend on . The projection B of B; in L =qL is also independent of , and (L; B ) is a crystal basis of W. Moreover for (j 0) 2 B; and ( j 0) 2 B; , we have
(j0) ( j 0) mod qL () P (u ) = P (u ) and P (u0 ) = P (u 0 ) : Finally, the crystal graph attached to (L; B ) is the bi-coloured graph given by il ( j0) () fiy(j0) = ( j0) () f^iu = u ; (j0) ?! ir (j 0) () fi(j0) = (j 0) () f^iu0 = u 0 : (j0) ?!
Proof: We imitate the proof of Theorem 4.3 and introduce the K (q)-linear map :
V
(11 )
V r V 1 V (c cr ) (d dr )
(1
1
)
(1
)
?! W ?! (c cr jd dr )
(1r )
1
1
1
Here ci; dj denote elements of the canonical bases of V i , V j , i.e. columns of size i , j , and (c cr jd dr ) is the quantum bitabloid formed on these columns. The map is in fact a homomorphism of Uq (gl n)-bimodules, and therefore sends the crystal lattice L; spanned by the tensors (c cr ) (d dr ) onto a crystal lattice L; in W. To describe precisely L; , it is enough to determine its submodule L; of highest weight vectors. We shall prove that L; = A (y jy ). To do this we have to introduce some notations. We write for short V := V 1
V r , and we consider a source vertex (1
1
)
(1
)
1
1
1
+
+
(1
(1
)
yk?r y . y = .. ... y1 yk 1
20
+1
)
of the crystal graph of V . This means that y yk is a Yamanouchi word on f1; : : : ; ng such that the columns of y are increasing from bottom to top. By de nition of a crystal basis, there exists a highest weight vector Ty in V such that Ty y mod qL; . Now, let 1
wk?r w . . w = . ... w1 wk 1
+1
be any monomial tensor of the same weight as y. Then (Ty w) = y;w (q) (y jy ) for some scalar y;w (q) 2 K (q). Indeed,
eyi (Ty w) = (eyi Ty w) = ((eiTy ) w) = 0 ; and on the other hand (Ty w) has weight (; ). Therefore (Ty w) is a scalar multiple of the highest weight vector (y jy ). But from the de nition of , we have (Ty w) =
X
u
u(q) (ujw) :
(33)
Here w is the tabloid made of the columns of w, u runs through the tabloids of the same shape and weight as w, and u(q) 2 A. Moreover, y (0) = 1 and u(0) = 0 for u 6= y. Expanding (33) on the basis
w = ftu1 w1 tuk wk j u uk has weight and wi = wj =) ui uj g 1
and comparing to the similar expansion of (y jy ), we obtain by checking the coecient of tw1 w1 twk wk that y;w (q) = w (q). Therefore, (Ty w) w y (y jy ) mod q(y jy ), and if y0 is another source vertex of ?(V ), we have (Ty Ty0 ) y y0 (y jy ) mod q(y jy ) :
(34)
This proves that L; = A (y jy ), as stated. It follows that L; does not in fact depend on , and we may write L; = L. We can now consider the map induced by from L; =qL; to L=qL (we still denote it by ). By (34), it maps to zero all the connected components of the crystal graph of V V with source vertex y y0 such that y 6= y0. This means precisely that (j0) 0 mod qL whenever Q(u) 6= Q(u0 ). On the other hand, to each corresponds a unique source vertex y = y( ) such that Q(uy ) = . The vertices w w0 of the connected components of ?(V V ) with origin y( ) y( ), , span a subspace of L; =qL; isomorphic under to L=qL. Therefore (L; B; mod qL) is a crystal basis of W, and this implies that B; mod qL is independent of . The other statements follow now easily from Theorem 3.8. 2 +
Proof of Theorem 1.1: Denote by L the crystal lattice of Fq [Matn ] whose submodule of highest weight vectors is equal to A (y jy ). It follows from the proof of Theorem 5.1 that the crystal lattice L of W is nothing but L \ W. We also see that L is spanned over A by the set of bitableaux ( j 0). Indeed, if we choose to be the standard
21
Young tableau whose rst column contains 1; 2; : : : ; 0 , whose second column contains 0 + 1; : : : ; 0 + 0 , and so on, we have 1
2
1
1
BT := f( j 0) j ; 0 are Young tableaux of shape g B;
id
and BT mod L is the part of B which labels the connected component of ?(W) corresponding to V V. Therefore, (L; tBT mod L) is a crystal basis of Fq[Matn] and L = A BT. On the other hand, taking = (k) in Theorem 5.1, we nd that the set of monomials
Bk = fti1j1 tik jk j Q(i ik ) = Q(j jk ) = for some g 1
1
gives rise to the same crystal basis under the guise (L; tk Bk mod L). Hence, applying again Theorem 3.4, it follows from the description of the crystal graph that mod qL if Q(i ik ) = Q(j jk ) (35) ti1j1 tik jk (P (i ik )0jP (j jk )) mod qL otherwise Finally, it is known that the coecients of the straightening of a bitabloid belong to K [q; q? ]. Since we have just proved that they also belong to A, we can replace in (35) the crystal lattice L by the K [q]-lattice L spanned by the quantum bitableaux. 2
1
1
1
1
1
References
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